Years
>
Year 12>
Trigonometric Functions>
Derivative of a Trigonometric Function>
Derivative of a Trigonometric Function 1Derivative of a Trigonometric Function 1
Try VividMath Premium to unlock full access
Time limit: 0
Quiz summary
0 of 6 questions completed
Questions:
- 1
- 2
- 3
- 4
- 5
- 6
Information
–
You have already completed the quiz before. Hence you can not start it again.
Quiz is loading...
You must sign in or sign up to start the quiz.
You have to finish following quiz, to start this quiz:
Loading...
- 1
- 2
- 3
- 4
- 5
- 6
- Answered
- Review
-
Question 1 of 6
1. Question
Find the derivative using the chain rule`5cos4x`Hint
Help VideoCorrect
Great Work!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Chain Rule
$$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$First, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$5\;\text{cos}\;\color{#9a00c7}{4x}$$ `x` `=` `4x` `n` `=` `1` Finally, substitute the values into the chain rule`y’` `=` $$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$ `=` $$5\cdot\color{#e65021}{1}\cdot\text{cos}\;(\color{#004ec4}{4x})\cdot\color{#00880A}{f'(4x)}$$ Substitute known values `=` $$5\cdot(-\text{sin}\;4x)\cdot\color{#00880A}{4}$$ Differentiate the values `=` $$-20\;\text{sin}\;4x$$ `y’=-20 \text(sin) 4x` -
Question 2 of 6
2. Question
Find the derivative using the chain rule`tan(4x-2)`Hint
Help VideoCorrect
Correct!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Chain Rule
$$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$First, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{\text{tan}\;(4x-2)}$$ `x` `=` `(4x-2)` `n` `=` `1` Finally, substitute the values into the chain rule`y’` `=` $$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$ `=` $$\color{#e65021}{1}\cdot\text{tan}\;(\color{#004ec4}{4x-2})\cdot\color{#00880A}{f'(4x-2)}$$ Substitute known values `=` $$(\text{sec}^2\;(4x-2)\cdot\color{#00880A}{4}$$ Differentiate the values `=` $$4\;\text{sec}^2\;(4x-2)$$ Simplify `y’=4 \text(sec)^2 (4x-2)` -
Question 3 of 6
3. Question
Find the derivative using the chain rule`sqrt(cosx)`Hint
Help VideoCorrect
Keep Going!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Chain Rule
$$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$First, convert the surd into an exponent`sqrt(\text(cos)x)` `=` `\text(cos) x^(1/2)` Next, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{\text{cos}\;x}^{\color{#e65021}{\frac{1}{2}}}$$ `x` `=` `\text(cos) x` `n` `=` `1/2` Finally, substitute the values into the chain rule`y’` `=` $$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$ `=` $$\color{#e65021}{\frac{1}{2}}\cdot(\color{#004ec4}{\text{cos}\;x})^{\color{#e65021}{(\frac{1}{2})}-1}\cdot\color{#00880A}{f'(\text{cos}\;x)}$$ Substitute known values `=` $$\frac{1}{2}(\text{cos}\;x)^{-\frac{1}{2}}\cdot\color{#00880A}{-\text{sin}\;x}$$ Differentiate `\text(cos) x` `=` $$\frac{-\text{sin}\;x}{2(\text{cos}\;x)^{\frac{1}{2}}}$$ Reciprocate `\text(cos) x^(-1/2)` `=` $$\frac{-\text{sin}\;x}{2\sqrt{\text{cos}\;x}}$$ Convert the exponent into a surd `y’=(-\text(sin) x)/(2sqrt(\text(cos) x))` -
Question 4 of 6
4. Question
Find the derivative using the chain rule`sin^4x`Hint
Help VideoCorrect
Fantastic!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Chain Rule
$$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$The expression can also be written as`(\text(sin) x)^4` Next, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{\text{sin}\;x}^{\color{#e65021}{4}}$$ `x` `=` `\text(sin) x` `n` `=` `4` Finally, substitute the values into the chain rule`y’` `=` $$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$ `=` $$\color{#e65021}{4}\cdot(\color{#004ec4}{\text{sin}\;x})^{\color{#e65021}{4}-1}\cdot\color{#00880A}{f'(\text{sin}\;x)}$$ Substitute known values `=` $$4\;(\text{sin}\;x)^{3}\cdot\color{#00880A}{\text{cos}\;x}$$ Differentiate the values `=` $$4\;\text{sin}^3\;x\;\text{cos}\;x$$ Evaluate `y’=4 \text(sin)^3 x \text(cos) x` -
Question 5 of 6
5. Question
Find the derivative using the chain rule`cos^3 2x`Hint
Help VideoCorrect
Excellent!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Chain Rule
$$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$The expression can also be written as`(\text(cos) 2x)^3` Next, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$\color{#9a00c7}{\text{cos}\;2x}^{\color{#e65021}{3}}$$ `x` `=` `\text(cos) 2x` `n` `=` `3` Finally, substitute the values into the chain rule`y’` `=` $$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$ `=` $$\color{#e65021}{3}\cdot(\color{#004ec4}{\text{cos}\;2x})^{\color{#e65021}{3}-1}\cdot\color{#00880A}{f'(\text{cos}\;2x)}$$ Substitute known values `=` $$3\;(\text{cos}\;2x)^{2}\cdot\color{#00880A}{-\text{sin}\;2x\cdot2}$$ Differentiate the values `=` $$-6\;\text{cos}^2\;2x\;\text{sin}\;2x$$ Evaluate `y’=-6 \text(cos)^2 2x \text(sin) 2x` -
Question 6 of 6
6. Question
Find the derivative using the chain rule`3 sin(5x-3)`Hint
Help VideoCorrect
Nice Job!
Incorrect
Derivatives of Trigonometric Functions
`y'(\text(sin))=\text(cos)``y'(\text(cos))=-\text(sin)``y'(\text(tan))=\text(sec)^2`Chain Rule
$$y’=\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$First, identify the values of the function`f(x)` `=` $$\color{#9a00c7}{x}^{\color{#e65021}{n}}$$ `f(x)` `=` $$3\;\text{sin}\;\color{#9a00c7}{(5x-3)}$$ `x` `=` `5x-3` `n` `=` `1` Finally, substitute the values into the chain rule`y’` `=` $$\color{#e65021}{n}\cdot(\color{#004ec4}{f(x)})^{\color{#e65021}{n}-1}\cdot\color{#00880A}{f'(x)}$$ `=` $$3\cdot\color{#e65021}{1}\cdot\text{sin}\;(\color{#004ec4}{5x-3})\cdot\color{#00880A}{f'(5x-3)}$$ Substitute known values `=` $$3\;(\text{cos}\;(5x-3))\cdot5$$ Differentiate the values `=` $$15\;\text{cos}\;(5x-3)$$ Evaluate `y’=15 \text(cos)(5x-3)`
Quizzes
- Converting Angle Measures 1
- Converting Angle Measures 2
- Converting Angle Measures 3
- Finding the Central Angle in a Circle
- Finding Areas in a Circle
- Values on the Unit Circle
- Finding Missing Angles Using the Unit Circle
- Trigonometric Ratios in the Unit Circle
- Trig Exact Values 1
- Trig Exact Values 2
- Trig Equations
- Derivative of a Trigonometric Function 1
- Derivative of a Trigonometric Function 2
- Derivative of a Trigonometric Function 3
- Applications of Differentiation
- Integral of a Trigonometric Function 1
- Integral of a Trigonometric Function 2
- Applications of Integration
- Graphing Trigonometric Functions 1
- Graphing Trigonometric Functions 2
- Graphing Trigonometric Functions 3
- Graphing Trigonometric Functions 4